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International Journal of Differential Equations
Volume 2010 (2010), Article ID 846107, 16 pages
Research Article

On the Selection and Meaning of Variable Order Operators for Dynamic Modeling

School of Engineering, University of California, P.O. Box 2039, Merced, CA 95344, USA

Received 4 August 2009; Accepted 8 October 2009

Academic Editor: Nikolai Leonenko

Copyright © 2010 Lynnette E. S. Ramirez and Carlos F. M. Coimbra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order (VO) operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude that the order of differentiation of a single VO operator that represents the dynamics of a viscoelastic oscillator in stationary motion is a normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians ( 𝜋 ), so that the normalization constant is simply 2 / 𝜋 .